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IEEE International Symposium on Information Theory (ISIT)

Quantum Coding via Semidefinite Programming

Mario Berta, Francesco Borderi, Omar Fawzi, Volkher B. Scholz

Published Jul 1, 2019• 1 citation
Publisher

Abstract

We derive converging hierarchies of efficiently computable semidefinite programming outer bounds on the optimal fidelity for the transmission of quantum information over noisy quantum channels. Based on positive partial transpose conditions we give a sufficient criterion for the exact convergence at any given level of the hierarchies. The worst case convergence speed of our hierarchies is quantified via positive semidefinite representable outer approximations on the set of separable Choi states, which are based on novel finite de Finetti theorems for quantum channels.

DOI
10.1109/isit.2019.8849325

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We derive converging hierarchies of efficiently computable semidefinite programming outer bounds on the optimal fidelity for the transmission of quantum information over noisy quantum channels. Based on positive partial transpose conditions we give a sufficient criterion for the exact convergence at any given level of the hierarchies. The worst case convergence speed of our hierarchies is quantified via positive semidefinite representable outer approximations on the set of separable Choi states, which are based on novel finite de Finetti theorems for quantum channels.
We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[M(X⊗Y)]\mathrm{Tr}\big[M(X\otimes Y)\big]Tr[M(X⊗Y)], maximized with respect to semidefinite constraints on XXX and YYY. Applied to the problem of quantum error correction this gives hierarchies of efficiently computable outer bounds on the optimal fidelity for any message dimension and error model. The first level of our hierarchies corresponds to the non-signalling assisted fidelity previously studied by [Leung & Matthews, IEEE Trans.~Inf.~Theory 2015], and positive partial transpose constraints can be added and used to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo's asymptotic de Finetti theorem for quantum channels. Finally, our proof methods also allow us to answer an open question from [Brand\~ao & Harrow, STOC 2013] by improving the approximation factor of de Finetti theorems with no symmetry from O(dk/2)O(d^{k/2})O(dk/2) to poly(d,k)\mathrm{poly}(d,k)poly(d,k), where ddd denotes local dimension and kkk the number of copies.
Abstract We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[H(D⊗E)]{\mathrm {Tr}}\big [H(D\otimes E)\big ]Tr[H(D⊗E)] Tr [ H ( D ⊗ E ) ] , maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(dk/2)O(d^{k/2})O(dk/2) O ( d k / 2 ) to poly(d,k){\mathrm {poly}}(d,k)poly(d,k) poly ( d , k ) , where d denotes local dimension and k the number of copies.
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of εεε in poly(1/ε,input dimension)\mathrm{poly}(1/ε, \text{input dimension})poly(1/ε,input dimension) time.
Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of$\epsilon ininin\mathrm {poly}(1/\epsilon, \text {input dimension})time,comparedtothetime, compared to thetime,comparedtothe\exp (1/\epsilon, \text {input dimension})$running time required for direct computation.
We investigate the classical communication over quantum channels when assisted by no-signaling and positive-partial-transpose-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and the one-shot E-error capacity are formalized as semidefinite programs (SDPs). Based on this, we obtain improved SDP finite blocklength converse bounds of general quantum channels for entanglement-assisted codes and unassisted codes. Furthermore, we derive two SDP strong converse bounds for the classical capacity of general quantum channels: for any code with a rate exceeding either of the two bounds of the channel, the success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bounds, we derive an improved upper bound on the classical capacity of the amplitude damping channel. We also establish the strong converse property for the classical and private capacities of a new class of quantum channels. We finally study the zero-error setting and provide efficiently computable upper bounds on the one-shot zero-error capacity of a general quantum channel.